If an order optimal bound of \(T_\star \)  is not known, we can follow the arguments in [1]} that sets \(\tilde{r}(s,a)=r(s,a) - \frac{1}{K^{n}}\)  for some \(n\gg 1\) . By this, we can also remove the dependency on \(T_{\max }\) , with the price of an additional \(\frac{T_\star }{K^{n-1}}\)  regret.
